Abstract
Based on derivatives of a Gaussian density, the Gram–Charlier series is an infinite expansion. Its truncated series is often used in many fields to approximate probability density functions. Although the expansions are useful, there are constrained regions on the value of the cumulants (or moments) that admit a valid (nonnegative) probability density function. When the truncation order is low (just at fourth-order), the truncated Gram–Charlier density may be difficult to approximate an implied probability distribution as closely as possible, especially for distributions that are not sufficiently close to a normal distribution. One might increase the order after which the series is truncated until a perfect fit is achieved. However, the series expansion is usually truncated in the existing literature until the fourth-order term because it becomes difficult to find valid regions. This paper shows how the valid region of higher cumulants can be numerically implemented by the semi-definite algorithm, which ensures that a series truncated at a cumulant of arbitrary even order represents a valid probability density. We provide examples of two valid regions of the sixth and eighth Gram–Charlier densities (i.e., truncated at the sixth and eighth terms). Our analysis proves the fact that valid regions can be broadened with the higher-order expansions. Furthermore, the impact of higher cumulants on the valid regions has been shown.